Mathematical Economics
specialty
An Introduction in Prospect
Theory
by: Chen Yeh
In the analysis of decision-making under uncertainty the expected utility hypothesis, formulated by von Neumann
and Morgenstern (1944), has been the dominating framework. It has in general been accepted as a normative model
of choice under risk (How should economic agents make decisions under risk?) and has also proved popular as a
descriptive model of economic behaviour (How do people make decisions in real life when faced with uncertainty?).
However the establishment of expected utility theory as a descriptive model has proved to be less successful and has
led to several critiques. One of the most cited critiques is Prospect Theory proposed by Kahnemann and Tversky
(1979). In this article, their intriguing framework is analyzed at the introductory level.
Introduction
The axioms of expected utility theory
Decision-making (under certainty) by the economic
agent has been one of the cornerstones of modern
microeconomics. Currently there is a theory that is based
on a set of axioms of rational consumer preferences and
has been standing firm for quite a while. A logical next
step for economic researchers was of course to come up
with a theory of decision-making under uncertainty. The
famous expected utility hypothesis was presented by von
Neumann and Morgenstern in 1944.
Although their theory from a normative perspective
has been considered as the leading paradigm in decisionmaking under uncertainty (or risk), their framework is
subject to a few paradoxes and has therefore lead to
several critiques. Economists (and psychologists) were
especially criticizing the capabilities of the theory as
a descriptive model. If people in their daily lives were
making decisions as predicted by the expected utility
hypothesis, then surely their preferences should also
satisfy the axioms of this theory.
However Kahneman and Tversky (1979, henceforth
K&T) showed in laboratory experiments that the
preferences of economic agents often violate the axioms
of expected utility theory. As a result, they presented their
own descriptive framework of decision-making under
risk: Prospect Theory. This article was published in the
prestigious journal Econometrica and has become one of
the most cited articles of this scientific magazine.
Decision-making under risk can be viewed as choice
between prospects (gambles or lotteries). A prospect
can be seen as a set of n outcomes, which each have a
certain probability. This is denoted mathematically by
(x1,p1;x2,p2;..; xn,pn). Since we are facing probabilities, the
sum of these probabilities must furthermore equal unity,
i.e. p1 + p2 +...+ pn = 1. The outcome of each prospect
gives the decision-maker or economic agent a certain
satisfaction or utility.
In expected utility theory, the following three
assumptions are made:
1. (Expectation)
U(x1, p1; x2, p2 ;..; xn, pn) = p1u(x1) + p2u(x2) +...+ pnu(xn)
Thus the overall utility of a prospect U is a weighted
average of the utility u of its outcomes.1
2. (Integration) Integrating an outcome w into a prospect
is acceptable if and only if:
U(w + x1, p1; w + x2 , p2 ;..; w + xn, pn) > u(w)
The above formula simply means that a decision-maker
is willing to bet with w in a lottery (i.e. w is integrated
into the lottery) if and only if the expected utility of this
In this issue of AENORM, we continue to present a series of articles. These series contain summaries of articles which have
been of great importance in economics or have caused considerable attention, be it in a positive sense or a controversial way.
Reading papers from scientific journals can be quite a demanding task for the beginning economist or econometrician. By
summarizing the selected articles in an understanding way, the AENORM sets its goal to reach these students in particular
and introduce them into the world of economic academics.
For questions or criticism, feel free to contact the AENORM editorial board at [email protected]
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AENORM vol. 17 (66) December 2009
Mathematical Economics
Figure 1. Illustration of Allais’ paradox (Kahneman and
Tversky, 1979)
lottery exceeds the utility of a certain outcome w. Thus
according to expected utility theory, people only seem
to care about the utility levels they end up with (final
states) and do not think in terms of utility gains or losses
(differences between states).
3. (Risk aversion) The utility function u is concave.
The economic interpretation of concavity is that a
person prefers to receive x with certainty to any risky
prospect that has expected value of x, thus a decisionmaker in general does not like risk.
Figure 2: Overweighting of miniscule probabilities (Kahneman and Tversky, 1979)
In the following set of problems (Figure 2), K&T
show how the substitution axiom of utility theory is
systematically violated.
In the first problem, people often tend to choose the
prospect where winning is more likely (i.e. B). In the
second part, the problem is structurally the same, but the
probabilities of winning are now miniscule. However
people now change their choice to C. Thus in line with
previous results, K&T conclude that people’s attitudes
towards risk are changing as function of probabilities. A
phenomenon that cannot be explained by expected utility
People tend to think in differences in wealth rather than
final states of wealth
Evidence from the laboratory: violations of
axioms and effects
According to expected utility theory, the utilities of
outcomes are weighted according to their respective
probabilities. However in a series of experiments, K&T
show that this axiom is systematically violated: people
often overweight outcomes that are considered as a sure
bet, relative to outcomes that are merely probable. In
their paper, K&T label this effect as the certainty effect.
In the following pair of problems (Figure 1), subjects
were asked to make a choice in each problem of monetary
outcomes. Where the asterisk indicates that the preference
was significant at the 1 percent level. It is clear that the
combination (B,C) was the most popular. Individual
patterns of choice indicated that a 61 percent majority
made this modal choice. However this contradicts
expected utility theory. To see why, note that problem 2
is obtained by simply eliminating a 66 percent chance
of winning 2400 from both prospects. Thus choice A
(B) and C (D) are equivalent according to the expected
utility hypothesis, but subjects seem to be inconsistent in
their choices. This paradox has been labeled as the Allais
paradox of expected utility theory.
theory.
In the previous pair of problems, K&T only used
positive prospects, i.e. people could only win money.
However K&T demonstrated that people’s attitudes
towards risk also change when they are faced with
negative prospects.
Table 1 illustrates that preferences between negative
prospects are the exact mirror image of the preference
between positive prospects. K&T termed this as the
reflection effect. The main implication of these results is
that the overweighting of certainty increases the aversion
of losses as well as the desirability of gains.
To resolve this problem, it was suggested that people
prefer prospects that have high expected value and
low variance. However Problem 4’ in table 1 seems to
contradict this conjecture: losing 3000 for sure has a
higher expected value and lower variance than losing
4000 with a 80 percent probability or losing nothing with
20 percent probability. Still, a highly significant majority
of 92 percent chose the latter.
In order to simplify choices, people often disregard
components that prospects share and focus on those parts
that distinguish prospects. However this may produce
preferences that are inconsistent as the following pair of
Note the difference between overall utility U and utility of one specific outcome u. The difference in use of capital letters is
subtle, but important to distinguish.
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Mathematical Economics
Table 1. Evaluation criteria for the three model specifications
Positive prospects
Problem 3
N = 95
Problem 4
N = 95
Problem 7
N = 66
Problem 8
N = 66
(4,000, 0.80)
[20]
(4,000, 0.20)
[65]*
(3,000, 0.90)
[86]*
(3,000, 0.002)
[27]
<
>
>
<
(3,000)
[80]*
(3,000, 0.25)
[35]
(6,000, 0.45)
[14]
(6,000, 0.001)
[73]*
problems will show (Figure 3).
In the two-stage version of the game (the third problem
above), there is a 0.25*0.80 = 0.20 chance to win 4000
and a 0.25*1.00 = 0.25 chance of winning 3000. Thus
this problem is equivalent to second problem above.
However the results of the two stage game were very
similar to those of the first problem above (contrary to
the prediction of expected utility theory). It seemed as if
people simply ignored the first stage of the game: K&T
describe this as the isolation effect.
Formalization of the framework: the editing and evaluation phases
The results of their experiments were a sufficient reason
for K&T to conclude that expected utility theory should
be disregarded as a descriptive model of choice behaviour
under risk. They coined their alternative as Prospect
Theory. This particular theory was originally developed
for simple prospects with monetary outcomes and stated
probabilities, but could be extended to more scenarios.
One important feature of prospect theory is the
division of the choice process into an editing phase and
an evaluation phase. In the editing phase, agents perform
Figure 3: The isolation effect (Kahneman and Tversky,
1979)
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AENORM vol. 17 (66) December 2009
Negative prospects
Problem 3’
N = 95
Problem 4’
N = 95
Problem 7’
N = 66
Problem 8’
N = 66
(-4,000, 0.80)
[92]*
(-4,000, 0.20)
[42]
(-3,000, 0.90)
[8]
(-3,000, 0.002)
[70]*
>
<
<
>
(-3,000)
[8]
(-3,000, 0.25)
[58]
(-6,000, 0.45)
[92]*
(-6,000, 0.001)
[30]
a preliminary analysis of the offered prospects, which
often results in simplifying the prospects. Consequently
in the evaluation phase, the edited prospects are evaluated
and the prospect with the highest value is chosen.
According to K&T, the function of the editing phase is
to reorganize and edit the prospects such that subsequent
evaluation and choice becomes easier for the subject in
question. K&T consider the following to be the major
operations:
- Coding: people perceive outcomes as gains and losses,
rather than final states of welfare. Thus people tend to
think in differences rather than final outcomes. Gains
and losses however have to be defined relative to some
situation. K&T call this the reference point.
- Combination: prospects can sometimes be simplified
by combining two prospects into one, e.g. winning
200 with 25 percent or winning 200 with 25 percent
can be simply reformulated as winning 200 with 50
percent.
- Segregation: prospects are often divided into riskless
and risky components, e.g. the prospect of winning 300
with 80 percent or 200 with 20 percent is decomposed
Figure 4: A hypothetical value function v (Kahneman and
Tversky, 1979)
Mathematical Economics
Figure 5: A hypothetical weighting function π (Kahneman
and Tversky, 1979)
weight. Therefore, K&T propose that probabilities should
be evaluated in the weighting function π(p), which results
in proper decision weights. However they explicitly
mention that these decision weights are not probabilities
as “decision weights measure the impact of events on
the desirability of events and not merely the perceived
likelihood of these events”. An example of a weighting
function relative to probabilities is given in the Figure 5.
Thus the value of a prospect in its whole is given by
V(x1, p1; x2, p2 ;..; xn, pn ; r)
= π(p1)v(x1 ; r) + π(p2)v(x2 ; r) +...+ π(pn)v(xn ; r)
into a sure gain of 200 and the prospect of winning
100 with 80 percent or nothing with 20 percent.
- Cancellation: people often disregard those components
that are shared by different prospects. This was
demonstrated in the isolation effect.
Formalization of the framework: the value
and weighting function
In the expected utility hypothesis, outcomes were
evaluated in utility functions. K&T suggested the use
of value functions. Two important features distinguish
the value function v from the classical utility function u.
First of all, people tend to think in differences in wealth
rather than final states of wealth. Second, these changes
in wealth are subject to the current state (or reference
point) of the decision-maker, e.g. the worth of winning
1000$, given that the person is a billionaire, is obviously
lower than the worth of winning this amount, given that
this person is a (poor) college student. Similarly, the
difference between losing 100$ or 200$ appears greater
as the difference between the loss of 1100$ and 1200$.2
Thus strictly speaking the value function has two
inputs: the impact of the outcome of the prospect relative
to the reference point and the magnitude of change in
wealth by the outcome of the prospect. The fact that v
is a function of the reference point seems to be a valid
reason for hypothesizing that v is concave above the
reference point and convex below it.3 Figure 4 shows the
functional form of a hypothetical value function.
The phenomenon of overweighting seems to imply that
probabilities alone are an insufficient tool as a decision
where r denotes the reference point. Although its
mathematical representation compared to expected utility
theory is subtle, the results are fundamentally different!
Preferences between risky options in prospect theory
are thus based on two principles. The first principle
concerns editing activities by the decision-maker that
determine how prospects are perceived. Consequently,
the second principle involves the subjective evaluation
of gains and losses and the weighting of uncertain
outcomes.
After several critical reaction papers, Kahneman and
Tversky (1992) revised their framework, which resulted
in Cumulative Prospect Theory. Their model has been
applied into many fields of economics and has proved to
be of valuable use. In 2002, Kahneman received the Nobel
Prize in Economic Sciences for integrating psychological
insights into economics, especially for his development
of Cumulative Prospect Theory.
References
Kahneman, D. and A. Tversky. “Prospect Theory: An
Analysis of Decision under Risk.” Econometrica, 47.2
(1979):263 – 291.
Kahneman, D. and A. Tversky. “Advances in Prospect
Theory: Cumulative Representation of Uncertainty.”
Journal of Risk and Uncertainty, 5 (1992):297 – 293.
The loss in the second scenario is relatively smaller for the decision-maker, although the loss in magnitude is the same as in
the first scenario.
3
For an intuitive notion of this concave/convex structure, note that it is easier to discriminate between a temperature change
from 3º to 6º than it is in a change from 13º to 16º.
2
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